document.write( "Question 1129620: prove n^3 + 2n is divisible by 3 induction \n" ); document.write( "
Algebra.Com's Answer #746195 by ikleyn(52784)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "The base of induction\r
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document.write( "    At n= 1  n^3 + 2n = 1^3 + 2*1 = 3  is divisible by 3.\r\n" );
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document.write( "    Thus the base of induction is valid.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The induction step\r
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document.write( "    Let assume that  P(n) = n^3 + 2n is divisible by 3,\r\n" );
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document.write( "    Then  P(n+1) = (n+1)^3 + 2*(n+1) = n^3 + 3n^2 + 3n + 1 + 2n + 2 = \r\n" );
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document.write( "                 = (re-group) = (n^3 + 2n) + (3n^2 + 3n + 3) = P(n) + 3*(n^2 + n +1).\r\n" );
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document.write( "    So, P(n+1) is the sum of P(N) and the other addend, which is multiple of 3.\r\n" );
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document.write( "    Thus, if P(n) is divisible by n, then P(n+1) is divisible by 3, too.\r\n" );
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document.write( "    The inductive step is proven to be valid.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Hence, according to the Mathematical induction principle, the statement is true for all positive integer n.\r
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\n" ); document.write( "\n" ); document.write( "The proof is completed.\r
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\n" ); document.write( "\n" ); document.write( "On the method of Mathematical induction see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction for sequences other than arithmetic or geometric\r
\n" ); document.write( "\n" ); document.write( "    - Proving inequalities by the method of Mathematical Induction\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on the Method of Mathematical induction\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this online textbook under the topic
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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